Toniolo and Linder, Equation (13)

Percentage Accurate: 50.0% → 64.3%

Time: 21.2s

Alternatives: 22

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot \left(U* - U\right), n \cdot \frac{l\_m}{Om}, \mathsf{fma}\left(l\_m \cdot -2, \frac{l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt
      (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
     (if (<= t_2 2e+306)
       (sqrt
        (*
         t_1
         (fma
          (* (/ l_m Om) (- U* U))
          (* n (/ l_m Om))
          (fma (* l_m -2.0) (/ l_m Om) t))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om Om)) (/ -2.0 Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * fma(((l_m / Om) * (U_42_ - U)), (n * (l_m / Om)), fma((l_m * -2.0), (l_m / Om), t))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(Float64(l_m / Om) * Float64(U_42_ - U)), Float64(n * Float64(l_m / Om)), fma(Float64(l_m * -2.0), Float64(l_m / Om), t))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 10.5

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 10.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 13.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 45.9%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 50.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 50.3%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 23.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

   7. Simplified 23.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, \color{blue}{t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{U* \cdot \ell}{Om}} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{U* \cdot \ell}{Om} \cdot \color{blue}{\frac{\ell \cdot n}{Om}}\right)} \]

      12. frac-times N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right)} \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      17. *-lowering-*.f64 17.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Applied egg-rr 17.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]

       10. sub-neg N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{U* \cdot n}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

       11. associate-/l* N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{U* \cdot \frac{n}{{Om}^{2}}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(U*, \frac{n}{{Om}^{2}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)}} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \color{blue}{\frac{n}{{Om}^{2}}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       16. associate-*r/ N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

       17. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\frac{\color{blue}{2}}{Om}\right)\right)} \]

       18. distribute-neg-frac N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{Om}}\right)} \]

       19. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{\color{blue}{-2}}{Om}\right)} \]

       20. /-lowering-/.f64 20.4

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{-2}{Om}}\right)} \]

   12. Simplified 20.4%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot \left(U* - U\right), n \cdot \frac{\ell}{Om}, \mathsf{fma}\left(\ell \cdot -2, \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;n \cdot \frac{\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{-Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(\left(l\_m \cdot l\_m\right) \cdot U*\right)\right)}{Om \cdot Om}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 2e+306)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (if (<= t_2 INFINITY)
         (* n (/ (* (* l_m (sqrt 2.0)) (sqrt (* U U*))) (- Om)))
         (sqrt (* (* n n) (/ (* 2.0 (* U (* (* l_m l_m) U*))) (* Om Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = n * (((l_m * sqrt(2.0)) * sqrt((U * U_42_))) / -Om);
	} else {
		tmp = sqrt(((n * n) * ((2.0 * (U * ((l_m * l_m) * U_42_))) / (Om * Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	elseif (t_2 <= Inf)
		tmp = Float64(n * Float64(Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * U_42_))) / Float64(-Om)));
	else
		tmp = sqrt(Float64(Float64(n * n) * Float64(Float64(2.0 * Float64(U * Float64(Float64(l_m * l_m) * U_42_))) / Float64(Om * Om))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(n * N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(n * n), $MachinePrecision] * N[(N[(2.0 * N[(U * N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 86.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 86.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 24.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 21.9%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   7. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U - U*}{Om \cdot Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \frac{U \cdot 2}{n}\right)} \cdot n} \]

   8. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{U \cdot \left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      5. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      7. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

      8. *-lowering-*.f64 9.9

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

   10. Simplified 9.9%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \cdot n \]

   11. Taylor expanded in l around -inf

       \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\right)\right)} \cdot n \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\right)\right)} \cdot n \]

       2. associate-*l/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{Om}}\right)\right) \cdot n \]

       3. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)}} \cdot n \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)}} \cdot n \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       6. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot U*}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       12. neg-lowering-neg.f64 20.1

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{\color{blue}{-Om}} \cdot n \]

   13. Simplified 20.1%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{-Om}} \cdot n \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 0.2%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   6. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\left(2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot {\ell}^{2}\right)\right)}{{Om}^{2}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot {\ell}^{2}\right)\right)}{{Om}^{2}}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{\color{blue}{2 \cdot \left(U \cdot \left(U* \cdot {\ell}^{2}\right)\right)}}{{Om}^{2}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \color{blue}{\left(U \cdot \left(U* \cdot {\ell}^{2}\right)\right)}}{{Om}^{2}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}\right)}{{Om}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{{Om}^{2}}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)}{{Om}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

      9. *-lowering-*.f64 33.4

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{\color{blue}{Om \cdot Om}}} \]

   8. Simplified 33.4%

      \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;n \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{-Om}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot n\right) \cdot \frac{2 \cdot \left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)\right)}{Om \cdot Om}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(\frac{l\_m}{Om} \cdot U*, n \cdot \frac{l\_m}{Om}, \mathsf{fma}\left(l\_m \cdot -2, \frac{l\_m}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt
      (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
     (if (<= t_2 2e+306)
       (sqrt
        (*
         t_1
         (fma
          (* (/ l_m Om) U*)
          (* n (/ l_m Om))
          (fma (* l_m -2.0) (/ l_m Om) t))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om Om)) (/ -2.0 Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * fma(((l_m / Om) * U_42_), (n * (l_m / Om)), fma((l_m * -2.0), (l_m / Om), t))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(Float64(l_m / Om) * U_42_), Float64(n * Float64(l_m / Om)), fma(Float64(l_m * -2.0), Float64(l_m / Om), t))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * N[(N[(N[(l$95$m / Om), $MachinePrecision] * U$42$), $MachinePrecision] * N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * -2.0), $MachinePrecision] * N[(l$95$m / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 10.5

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 10.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 13.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 45.9%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 50.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 50.3%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

   7. Simplified 99.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 23.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

   7. Simplified 23.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, \color{blue}{t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{U* \cdot \ell}{Om}} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{U* \cdot \ell}{Om} \cdot \color{blue}{\frac{\ell \cdot n}{Om}}\right)} \]

      12. frac-times N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right)} \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      17. *-lowering-*.f64 17.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Applied egg-rr 17.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]

       10. sub-neg N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{U* \cdot n}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

       11. associate-/l* N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{U* \cdot \frac{n}{{Om}^{2}}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(U*, \frac{n}{{Om}^{2}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)}} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \color{blue}{\frac{n}{{Om}^{2}}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       16. associate-*r/ N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

       17. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\frac{\color{blue}{2}}{Om}\right)\right)} \]

       18. distribute-neg-frac N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{Om}}\right)} \]

       19. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{\color{blue}{-2}}{Om}\right)} \]

       20. /-lowering-/.f64 20.4

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{-2}{Om}}\right)} \]

   12. Simplified 20.4%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\frac{\ell}{Om} \cdot U*, n \cdot \frac{\ell}{Om}, \mathsf{fma}\left(\ell \cdot -2, \frac{\ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t + \frac{\left(n \cdot \frac{l\_m}{Om}\right) \cdot \left(l\_m \cdot \left(U* - U\right)\right) - 2 \cdot \left(l\_m \cdot l\_m\right)}{Om}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* U (* 2.0 n))
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_1 4e-318)
     (sqrt
      (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
     (if (<= t_1 2e+306)
       (sqrt
        (*
         (+
          t
          (/ (- (* (* n (/ l_m Om)) (* l_m (- U* U))) (* 2.0 (* l_m l_m))) Om))
         (* n (* 2.0 U))))
       (*
        (* l_m (sqrt 2.0))
        (sqrt (* (* n U) (fma U* (/ n (* Om Om)) (/ -2.0 Om)))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (U * (2.0 * n)) * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_1 <= 4e-318) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (t_1 <= 2e+306) {
		tmp = sqrt(((t + ((((n * (l_m / Om)) * (l_m * (U_42_ - U))) - (2.0 * (l_m * l_m))) / Om)) * (n * (2.0 * U))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_1 <= 4e-318)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (t_1 <= 2e+306)
		tmp = sqrt(Float64(Float64(t + Float64(Float64(Float64(Float64(n * Float64(l_m / Om)) * Float64(l_m * Float64(U_42_ - U))) - Float64(2.0 * Float64(l_m * l_m))) / Om)) * Float64(n * Float64(2.0 * U))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-318], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+306], N[Sqrt[N[(N[(t + N[(N[(N[(N[(n * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 10.5

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 10.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 13.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 45.9%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 50.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 50.3%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]
   5. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

   6. Applied egg-rr 94.4%

      \[\leadsto \color{blue}{\sqrt{\left(t - \frac{2 \cdot \left(\ell \cdot \ell\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(-\left(U - U*\right) \cdot \ell\right)}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 23.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

   7. Simplified 23.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, \color{blue}{t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{U* \cdot \ell}{Om}} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{U* \cdot \ell}{Om} \cdot \color{blue}{\frac{\ell \cdot n}{Om}}\right)} \]

      12. frac-times N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right)} \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      17. *-lowering-*.f64 17.2

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Applied egg-rr 17.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]

       10. sub-neg N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{U* \cdot n}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

       11. associate-/l* N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{U* \cdot \frac{n}{{Om}^{2}}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(U*, \frac{n}{{Om}^{2}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)}} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \color{blue}{\frac{n}{{Om}^{2}}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       16. associate-*r/ N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

       17. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\frac{\color{blue}{2}}{Om}\right)\right)} \]

       18. distribute-neg-frac N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{Om}}\right)} \]

       19. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{\color{blue}{-2}}{Om}\right)} \]

       20. /-lowering-/.f64 20.4

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{-2}{Om}}\right)} \]

   12. Simplified 20.4%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(t + \frac{\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\ell \cdot \left(U* - U\right)\right) - 2 \cdot \left(\ell \cdot \ell\right)}{Om}\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m \cdot \left(n \cdot l\_m\right)}{Om \cdot Om}, U*, t\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-159)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 5e+144)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (sqrt
        (* U (* (* 2.0 n) (fma (/ (* l_m (* n l_m)) (* Om Om)) U* t))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-159) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 5e+144) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((U * ((2.0 * n) * fma(((l_m * (n * l_m)) / (Om * Om)), U_42_, t))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-159)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 5e+144)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(Float64(l_m * Float64(n * l_m)) / Float64(Om * Om)), U_42_, t))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-159], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 5e+144], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * U$42$ + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999998e-159

   1. Initial program 12.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 43.9

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 43.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 1.99999999999999998e-159 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 4.9999999999999999e144

   1. Initial program 97.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 87.0

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 87.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if 4.9999999999999999e144 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 18.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 18.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 18.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 28.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 31.5%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   11. Taylor expanded in U around 0

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \color{blue}{U*}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   12. Step-by-step derivation

   13. Simplified 31.6%

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \color{blue}{U*}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om \cdot Om}, U*, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{-Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))))
   (if (<= t_2 2e-159)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 2e+153)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (* n (/ (* (* l_m (sqrt 2.0)) (sqrt (* U U*))) (- Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double tmp;
	if (t_2 <= 2e-159) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+153) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = n * (((l_m * sqrt(2.0)) * sqrt((U * U_42_))) / -Om);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_2 <= 2e-159)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 2e+153)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = Float64(n * Float64(Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(U * U_42_))) / Float64(-Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-159], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+153], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-Om)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999998e-159

   1. Initial program 12.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 43.9

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 43.9%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 1.99999999999999998e-159 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2e153

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 86.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 86.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if 2e153 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 16.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 14.8%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   7. Applied egg-rr 11.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U - U*}{Om \cdot Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \frac{U \cdot 2}{n}\right)} \cdot n} \]

   8. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{U \cdot \left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      5. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      7. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

      8. *-lowering-*.f64 13.0

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

   10. Simplified 13.0%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \cdot n \]

   11. Taylor expanded in l around -inf

       \[\leadsto \color{blue}{\left(-1 \cdot \left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\right)\right)} \cdot n \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\ell \cdot \sqrt{2}}{Om} \cdot \sqrt{U \cdot U*}\right)\right)} \cdot n \]

       2. associate-*l/ N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{Om}}\right)\right) \cdot n \]

       3. distribute-neg-frac2 N/A

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)}} \cdot n \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)}} \cdot n \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       6. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot U*}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot U*}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       10. *-commutative N/A

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{\mathsf{neg}\left(Om\right)} \cdot n \]

       12. neg-lowering-neg.f64 23.4

           \[\leadsto \frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{\color{blue}{-Om}} \cdot n \]

   13. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{U* \cdot U}}{-Om}} \cdot n \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot U*}}{-Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 2e+306)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (/ (* (sqrt (* U U*)) (* l_m (* n (sqrt 2.0)))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = (sqrt((U * U_42_)) * (l_m * (n * sqrt(2.0)))) / Om;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = Float64(Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(n * sqrt(2.0)))) / Om);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 86.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 86.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 23.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U* around inf

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\ell \cdot \color{blue}{\frac{n \cdot \sqrt{2}}{Om}}\right) \cdot \sqrt{U \cdot U*} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\color{blue}{n \cdot \sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \color{blue}{\sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \color{blue}{\sqrt{U \cdot U*}} \]

      8. *-lowering-*.f64 19.2

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{\color{blue}{U \cdot U*}} \]

   7. Simplified 19.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}} \cdot \sqrt{U \cdot U*} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U \cdot U*}}{Om}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U \cdot U*}}{Om}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U \cdot U*}}}{Om} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)} \cdot \sqrt{U \cdot U*}}{Om} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \color{blue}{\left(n \cdot \sqrt{2}\right)}\right) \cdot \sqrt{U \cdot U*}}{Om} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \left(n \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \sqrt{U \cdot U*}}{Om} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\sqrt{U \cdot U*}}}{Om} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{Om} \]

      10. *-lowering-*.f64 21.7

          \[\leadsto \frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{\color{blue}{U* \cdot U}}}{Om} \]

   9. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right) \cdot \sqrt{U* \cdot U}}{Om}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{U \cdot U*} \cdot \left(\ell \cdot \left(n \cdot \sqrt{2}\right)\right)}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot l\_m\right) \cdot \frac{\sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 2e+306)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (* (sqrt (* U U*)) (* (* n l_m) (/ (sqrt 2.0) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((U * U_42_)) * ((n * l_m) * (sqrt(2.0) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(Float64(n * l_m) * Float64(sqrt(2.0) / Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(N[(n * l$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 86.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 86.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 23.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U* around inf

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\ell \cdot \color{blue}{\frac{n \cdot \sqrt{2}}{Om}}\right) \cdot \sqrt{U \cdot U*} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\color{blue}{n \cdot \sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \color{blue}{\sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \color{blue}{\sqrt{U \cdot U*}} \]

      8. *-lowering-*.f64 19.2

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{\color{blue}{U \cdot U*}} \]

   7. Simplified 19.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(\ell \cdot \color{blue}{\left(n \cdot \frac{\sqrt{2}}{Om}\right)}\right) \cdot \sqrt{U \cdot U*} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\ell \cdot n\right) \cdot \frac{\sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\ell \cdot n\right) \cdot \frac{\sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\ell \cdot n\right)} \cdot \frac{\sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \left(\left(\ell \cdot n\right) \cdot \color{blue}{\frac{\sqrt{2}}{Om}}\right) \cdot \sqrt{U \cdot U*} \]

      6. sqrt-lowering-sqrt.f64 21.4

         \[\leadsto \left(\left(\ell \cdot n\right) \cdot \frac{\color{blue}{\sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

   9. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{\left(\left(\ell \cdot n\right) \cdot \frac{\sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(l\_m \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (* (sqrt (* U U*)) (* l_m (/ (* n (sqrt 2.0)) Om)))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = sqrt((U * U_42_)) * (l_m * ((n * sqrt(2.0)) / Om));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = Float64(sqrt(Float64(U * U_42_)) * Float64(l_m * Float64(Float64(n * sqrt(2.0)) / Om)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U * U$42$), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(N[(n * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 60.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 51.8

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 0.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U* around inf

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)} \cdot \sqrt{U \cdot U*} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \left(\ell \cdot \color{blue}{\frac{n \cdot \sqrt{2}}{Om}}\right) \cdot \sqrt{U \cdot U*} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{\color{blue}{n \cdot \sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \color{blue}{\sqrt{2}}}{Om}\right) \cdot \sqrt{U \cdot U*} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \color{blue}{\sqrt{U \cdot U*}} \]

      8. *-lowering-*.f64 18.1

         \[\leadsto \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{\color{blue}{U \cdot U*}} \]

   7. Simplified 18.1%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right) \cdot \sqrt{U \cdot U*}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot U*} \cdot \left(\ell \cdot \frac{n \cdot \sqrt{2}}{Om}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(l\_m \cdot l\_m, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot U*\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma (* l_m l_m) (/ -2.0 Om) t)))
       (* n (/ (sqrt (* (* 2.0 U) (* (* l_m l_m) U*))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma((l_m * l_m), (-2.0 / Om), t)));
	} else {
		tmp = n * (sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(Float64(l_m * l_m), Float64(-2.0 / Om), t)));
	else
		tmp = Float64(n * Float64(sqrt(Float64(Float64(2.0 * U) * Float64(Float64(l_m * l_m) * U_42_))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(-2.0 / Om), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < +inf.0

   1. Initial program 60.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}} + t\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\frac{\color{blue}{{\ell}^{2} \cdot -2}}{Om} + t\right)} \]

      4. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{{\ell}^{2} \cdot \frac{-2}{Om}} + t\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{-2}{Om}, t\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{-2}{Om}, t\right)} \]

      8. /-lowering-/.f64 51.8

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \color{blue}{\frac{-2}{Om}}, t\right)} \]

   5. Simplified 51.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 0.2%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   7. Applied egg-rr 0.4%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U - U*}{Om \cdot Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \frac{U \cdot 2}{n}\right)} \cdot n} \]

   8. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{U \cdot \left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      5. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      7. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

      8. *-lowering-*.f64 21.2

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

   10. Simplified 21.2%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \cdot n \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}} \cdot n} \]

       2. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}} \cdot n \]

       3. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{Om \cdot Om}}} \cdot n \]

       4. pow2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{\color{blue}{{Om}^{2}}}} \cdot n \]

       5. sqrt-pow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \cdot n \]

       6. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{{Om}^{\color{blue}{1}}} \cdot n \]

       7. unpow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{Om}} \cdot n \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{Om}} \cdot n \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}}{Om} \cdot n \]

       10. associate-*r* N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       14. *-lowering-*.f64 21.6

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}}{Om} \cdot n \]

   12. Applied egg-rr 21.6%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot U*\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (*
          t_1
          (+
           (- t (* 2.0 (/ (* l_m l_m) Om)))
           (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
   (if (<= t_2 4e-318)
     (sqrt (* (* 2.0 U) (* n t)))
     (if (<= t_2 2e+306)
       (sqrt (* t_1 t))
       (* n (/ (sqrt (* (* 2.0 U) (* (* l_m l_m) U*))) Om))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+306) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = n * (sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = t_1 * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))
    if (t_2 <= 4d-318) then
        tmp = sqrt(((2.0d0 * u) * (n * t)))
    else if (t_2 <= 2d+306) then
        tmp = sqrt((t_1 * t))
    else
        tmp = n * (sqrt(((2.0d0 * u) * ((l_m * l_m) * u_42))) / om)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)));
	double tmp;
	if (t_2 <= 4e-318) {
		tmp = Math.sqrt(((2.0 * U) * (n * t)));
	} else if (t_2 <= 2e+306) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = n * (Math.sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))
	tmp = 0
	if t_2 <= 4e-318:
		tmp = math.sqrt(((2.0 * U) * (n * t)))
	elif t_2 <= 2e+306:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = n * (math.sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om)
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_2 <= 4e-318)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = Float64(n * Float64(sqrt(Float64(Float64(2.0 * U) * Float64(Float64(l_m * l_m) * U_42_))) / Om));
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)));
	tmp = 0.0;
	if (t_2 <= 4e-318)
		tmp = sqrt(((2.0 * U) * (n * t)));
	elseif (t_2 <= 2e+306)
		tmp = sqrt((t_1 * t));
	else
		tmp = n * (sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-318], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+306], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 3.9999999e-318

   1. Initial program 11.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 41.7

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 41.7%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 3.9999999e-318 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.00000000000000003e306

   1. Initial program 97.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   4. Step-by-step derivation

   5. Simplified 78.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]

   if 2.00000000000000003e306 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 17.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 15.2%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   7. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U - U*}{Om \cdot Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \frac{U \cdot 2}{n}\right)} \cdot n} \]

   8. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{U \cdot \left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      5. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      7. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

      8. *-lowering-*.f64 13.3

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

   10. Simplified 13.3%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \cdot n \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}} \cdot n} \]

       2. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}} \cdot n \]

       3. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{Om \cdot Om}}} \cdot n \]

       4. pow2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{\color{blue}{{Om}^{2}}}} \cdot n \]

       5. sqrt-pow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \cdot n \]

       6. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{{Om}^{\color{blue}{1}}} \cdot n \]

       7. unpow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{Om}} \cdot n \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{Om}} \cdot n \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}}{Om} \cdot n \]

       10. associate-*r* N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       14. *-lowering-*.f64 15.9

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}}{Om} \cdot n \]

   12. Applied egg-rr 15.9%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 4 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := U \cdot \left(2 \cdot n\right)\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{l\_m \cdot l\_m}{Om}\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)}\\ t_3 := \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-159}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{t\_1 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (* U (* 2.0 n)))
        (t_2
         (sqrt
          (*
           t_1
           (+
            (- t (* 2.0 (/ (* l_m l_m) Om)))
            (* (* n (pow (/ l_m Om) 2.0)) (- U* U))))))
        (t_3 (sqrt (* (* 2.0 U) (* n t)))))
   (if (<= t_2 2e-159) t_3 (if (<= t_2 2e+151) (sqrt (* t_1 t)) t_3))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double t_3 = sqrt(((2.0 * U) * (n * t)));
	double tmp;
	if (t_2 <= 2e-159) {
		tmp = t_3;
	} else if (t_2 <= 2e+151) {
		tmp = sqrt((t_1 * t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = u * (2.0d0 * n)
    t_2 = sqrt((t_1 * ((t - (2.0d0 * ((l_m * l_m) / om))) + ((n * ((l_m / om) ** 2.0d0)) * (u_42 - u)))))
    t_3 = sqrt(((2.0d0 * u) * (n * t)))
    if (t_2 <= 2d-159) then
        tmp = t_3
    else if (t_2 <= 2d+151) then
        tmp = sqrt((t_1 * t))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = U * (2.0 * n);
	double t_2 = Math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * Math.pow((l_m / Om), 2.0)) * (U_42_ - U)))));
	double t_3 = Math.sqrt(((2.0 * U) * (n * t)));
	double tmp;
	if (t_2 <= 2e-159) {
		tmp = t_3;
	} else if (t_2 <= 2e+151) {
		tmp = Math.sqrt((t_1 * t));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	t_1 = U * (2.0 * n)
	t_2 = math.sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * math.pow((l_m / Om), 2.0)) * (U_42_ - U)))))
	t_3 = math.sqrt(((2.0 * U) * (n * t)))
	tmp = 0
	if t_2 <= 2e-159:
		tmp = t_3
	elif t_2 <= 2e+151:
		tmp = math.sqrt((t_1 * t))
	else:
		tmp = t_3
	return tmp

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(U * Float64(2.0 * n))
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l_m * l_m) / Om))) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U)))))
	t_3 = sqrt(Float64(Float64(2.0 * U) * Float64(n * t)))
	tmp = 0.0
	if (t_2 <= 2e-159)
		tmp = t_3;
	elseif (t_2 <= 2e+151)
		tmp = sqrt(Float64(t_1 * t));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(n, U, t, l_m, Om, U_42_)
	t_1 = U * (2.0 * n);
	t_2 = sqrt((t_1 * ((t - (2.0 * ((l_m * l_m) / Om))) + ((n * ((l_m / Om) ^ 2.0)) * (U_42_ - U)))));
	t_3 = sqrt(((2.0 * U) * (n * t)));
	tmp = 0.0;
	if (t_2 <= 2e-159)
		tmp = t_3;
	elseif (t_2 <= 2e+151)
		tmp = sqrt((t_1 * t));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 2e-159], t$95$3, If[LessEqual[t$95$2, 2e+151], N[Sqrt[N[(t$95$1 * t), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1.99999999999999998e-159 or 2.00000000000000003e151 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 16.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

      4. *-lowering-*.f64 18.4

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

   5. Simplified 18.4%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   if 1.99999999999999998e-159 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 2.00000000000000003e151

   1. Initial program 97.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
   4. Step-by-step derivation

   5. Simplified 78.4%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m \cdot l\_m}{Om}\\ \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot t\_1\right) + \left(n \cdot {\left(\frac{l\_m}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, t\_1, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(l\_m \cdot l\_m\right) \cdot U*\right)}}{Om}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ (* l_m l_m) Om)))
   (if (<=
        (sqrt
         (*
          (* U (* 2.0 n))
          (+ (- t (* 2.0 t_1)) (* (* n (pow (/ l_m Om) 2.0)) (- U* U)))))
        INFINITY)
     (sqrt (* (* 2.0 U) (* n (fma -2.0 t_1 t))))
     (* n (/ (sqrt (* (* 2.0 U) (* (* l_m l_m) U*))) Om)))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = (l_m * l_m) / Om;
	double tmp;
	if (sqrt(((U * (2.0 * n)) * ((t - (2.0 * t_1)) + ((n * pow((l_m / Om), 2.0)) * (U_42_ - U))))) <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * U) * (n * fma(-2.0, t_1, t))));
	} else {
		tmp = n * (sqrt(((2.0 * U) * ((l_m * l_m) * U_42_))) / Om);
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(Float64(l_m * l_m) / Om)
	tmp = 0.0
	if (sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(Float64(t - Float64(2.0 * t_1)) + Float64(Float64(n * (Float64(l_m / Om) ^ 2.0)) * Float64(U_42_ - U))))) <= Inf)
		tmp = sqrt(Float64(Float64(2.0 * U) * Float64(n * fma(-2.0, t_1, t))));
	else
		tmp = Float64(n * Float64(sqrt(Float64(Float64(2.0 * U) * Float64(Float64(l_m * l_m) * U_42_))) / Om));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(N[(t - N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(n * N[Power[N[(l$95$m / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], Infinity], N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * N[(-2.0 * t$95$1 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(n * N[(N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] * U$42$), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 53.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 58.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)}\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right)}\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

      9. *-lowering-*.f64 47.1

         \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\ell \cdot \ell}}{Om}, t\right)\right)} \]

   7. Simplified 47.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 0.3%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   7. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \left(U \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{U - U*}{Om \cdot Om}, \mathsf{fma}\left(\ell, \frac{\ell}{Om \cdot -0.5}, t\right) \cdot \frac{U \cdot 2}{n}\right)} \cdot n} \]

   8. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{U \cdot \left(U* \cdot {\ell}^{2}\right)}{{Om}^{2}}}} \cdot n \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{U \cdot \left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \color{blue}{\left(U* \cdot {\ell}^{2}\right)}}{{Om}^{2}}} \cdot n \]

      5. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{Om}^{2}}} \cdot n \]

      7. unpow2 N/A

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

      8. *-lowering-*.f64 19.4

         \[\leadsto \sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{\color{blue}{Om \cdot Om}}} \cdot n \]

   10. Simplified 19.4%

       \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}} \cdot n \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}} \cdot n} \]

       2. associate-*r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}{Om \cdot Om}}} \cdot n \]

       3. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{Om \cdot Om}}} \cdot n \]

       4. pow2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\sqrt{\color{blue}{{Om}^{2}}}} \cdot n \]

       5. sqrt-pow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{{Om}^{\left(\frac{2}{2}\right)}}} \cdot n \]

       6. metadata-eval N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{{Om}^{\color{blue}{1}}} \cdot n \]

       7. unpow1 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{\color{blue}{Om}} \cdot n \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}{Om}} \cdot n \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(U \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)\right)}}}{Om} \cdot n \]

       10. associate-*r* N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(U* \cdot \left(\ell \cdot \ell\right)\right)}}}{Om} \cdot n \]

       14. *-lowering-*.f64 19.8

           \[\leadsto \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}}{Om} \cdot n \]

   12. Applied egg-rr 19.8%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot U\right) \cdot \left(U* \cdot \left(\ell \cdot \ell\right)\right)}}{Om} \cdot n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;n \cdot \frac{\sqrt{\left(2 \cdot U\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot U*\right)}}{Om}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.85 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.25e-29)
   (sqrt (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
   (if (<= l_m 1.85e+95)
     (sqrt
      (*
       (* U (* 2.0 n))
       (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
     (*
      (* l_m (sqrt 2.0))
      (sqrt (* (* n U) (- (/ (* n (- U* U)) (* Om Om)) (/ 2.0 Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.25e-29) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (l_m <= 1.85e+95) {
		tmp = sqrt(((U * (2.0 * n)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * (((n * (U_42_ - U)) / (Om * Om)) - (2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.25e-29)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (l_m <= 1.85e+95)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * Float64(Float64(Float64(n * Float64(U_42_ - U)) / Float64(Om * Om)) - Float64(2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.25e-29], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.85e+95], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(N[(N[(n * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] - N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.24999999999999996e-29

   1. Initial program 48.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 44.1

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 44.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 49.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 52.2%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 57.4

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 57.4%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 1.24999999999999996e-29 < l < 1.8500000000000001e95

   1. Initial program 45.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

   5. Simplified 51.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}} \]

   if 1.8500000000000001e95 < l

   1. Initial program 27.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 39.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{n \cdot \left(U* - U\right)}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{\frac{n \cdot \left(U* - U\right)}{{Om}^{2}}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{\color{blue}{n \cdot \left(U* - U\right)}}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \color{blue}{\left(U* - U\right)}}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{\color{blue}{Om \cdot Om}} - 2 \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2 \cdot 1}{Om}}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{\color{blue}{2}}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \color{blue}{\frac{2}{Om}}\right)} \cdot \left(\ell \cdot \sqrt{2}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \color{blue}{\left(\ell \cdot \sqrt{2}\right)} \]

   7. Simplified 60.5%

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.85 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \left(\frac{n \cdot \left(U* - U\right)}{Om \cdot Om} - \frac{2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.1 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.35 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;l\_m \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot l\_m\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, -4 \cdot \frac{U}{n \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 3.1e-27)
   (sqrt (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
   (if (<= l_m 1.35e+170)
     (sqrt
      (*
       (* U (* 2.0 n))
       (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
     (if (<= l_m 5e+226)
       (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
       (*
        (* n l_m)
        (sqrt
         (fma -2.0 (/ (* U (- U U*)) (* Om Om)) (* -4.0 (/ U (* n Om))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 3.1e-27) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (l_m <= 1.35e+170) {
		tmp = sqrt(((U * (2.0 * n)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else if (l_m <= 5e+226) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = (n * l_m) * sqrt(fma(-2.0, ((U * (U - U_42_)) / (Om * Om)), (-4.0 * (U / (n * Om)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 3.1e-27)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (l_m <= 1.35e+170)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	elseif (l_m <= 5e+226)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = Float64(Float64(n * l_m) * sqrt(fma(-2.0, Float64(Float64(U * Float64(U - U_42_)) / Float64(Om * Om)), Float64(-4.0 * Float64(U / Float64(n * Om))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 3.1e-27], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.35e+170], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 5e+226], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(n * l$95$m), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[(N[(U * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(U / N[(n * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < 3.0999999999999998e-27

   1. Initial program 47.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 43.9

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 43.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 48.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 51.9%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 57.1

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 57.1%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 3.0999999999999998e-27 < l < 1.3500000000000001e170

   1. Initial program 46.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

   5. Simplified 46.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}} \]

   if 1.3500000000000001e170 < l < 5.0000000000000005e226

   1. Initial program 20.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 22.2

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 22.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 40.4

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 40.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 49.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 49.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]

   if 5.0000000000000005e226 < l

   1. Initial program 14.4%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in n around inf

      \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{{n}^{2} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right)} \cdot \left(-2 \cdot \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left({\ell}^{2} \cdot \left(U - U*\right)\right)}{{Om}^{2}}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left(U \cdot {\ell}^{2}\right) \cdot \left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{\left({\ell}^{2} \cdot U\right)} \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot U\right) \cdot \left(U - U*\right)}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      13. unpow2 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, 2 \cdot \frac{U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}{n}\right)} \]

      15. associate-*r/ N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{\frac{2 \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}{n}}\right)} \]

   5. Simplified 12.9%

      \[\leadsto \sqrt{\color{blue}{\left(n \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\left(\left(\ell \cdot \ell\right) \cdot U\right) \cdot \left(U - U*\right)}{Om \cdot Om}, \frac{\left(2 \cdot U\right) \cdot \mathsf{fma}\left(\ell \cdot \ell, \frac{-2}{Om}, t\right)}{n}\right)}} \]

   6. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\left(\ell \cdot n\right) \cdot \sqrt{-4 \cdot \frac{U}{Om \cdot n} + -2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\ell \cdot n\right) \cdot \sqrt{-4 \cdot \frac{U}{Om \cdot n} + -2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\ell \cdot n\right)} \cdot \sqrt{-4 \cdot \frac{U}{Om \cdot n} + -2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}} \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \color{blue}{\sqrt{-4 \cdot \frac{U}{Om \cdot n} + -2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}} \]

      4. +-commutative N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\color{blue}{-2 \cdot \frac{U \cdot \left(U - U*\right)}{{Om}^{2}} + -4 \cdot \frac{U}{Om \cdot n}}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{{Om}^{2}}, -4 \cdot \frac{U}{Om \cdot n}\right)}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \color{blue}{\frac{U \cdot \left(U - U*\right)}{{Om}^{2}}}, -4 \cdot \frac{U}{Om \cdot n}\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{\color{blue}{U \cdot \left(U - U*\right)}}{{Om}^{2}}, -4 \cdot \frac{U}{Om \cdot n}\right)} \]

      8. --lowering--.f64 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \color{blue}{\left(U - U*\right)}}{{Om}^{2}}, -4 \cdot \frac{U}{Om \cdot n}\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, -4 \cdot \frac{U}{Om \cdot n}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{\color{blue}{Om \cdot Om}}, -4 \cdot \frac{U}{Om \cdot n}\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, \color{blue}{-4 \cdot \frac{U}{Om \cdot n}}\right)} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, -4 \cdot \color{blue}{\frac{U}{Om \cdot n}}\right)} \]

      13. *-lowering-*.f64 49.9

          \[\leadsto \left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, -4 \cdot \frac{U}{\color{blue}{Om \cdot n}}\right)} \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{\left(\ell \cdot n\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, -4 \cdot \frac{U}{Om \cdot n}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(n \cdot \ell\right) \cdot \sqrt{\mathsf{fma}\left(-2, \frac{U \cdot \left(U - U*\right)}{Om \cdot Om}, -4 \cdot \frac{U}{n \cdot Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;l\_m \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(l\_m \cdot l\_m\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= l_m 1.85e-28)
   (sqrt (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
   (if (<= l_m 1.8e+95)
     (sqrt
      (*
       (* U (* 2.0 n))
       (- t (/ (* (* l_m l_m) (fma n (/ (- U U*) Om) 2.0)) Om))))
     (*
      (* l_m (sqrt 2.0))
      (sqrt (* (* n U) (fma U* (/ n (* Om Om)) (/ -2.0 Om))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (l_m <= 1.85e-28) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (l_m <= 1.8e+95) {
		tmp = sqrt(((U * (2.0 * n)) * (t - (((l_m * l_m) * fma(n, ((U - U_42_) / Om), 2.0)) / Om))));
	} else {
		tmp = (l_m * sqrt(2.0)) * sqrt(((n * U) * fma(U_42_, (n / (Om * Om)), (-2.0 / Om))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (l_m <= 1.85e-28)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (l_m <= 1.8e+95)
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t - Float64(Float64(Float64(l_m * l_m) * fma(n, Float64(Float64(U - U_42_) / Om), 2.0)) / Om))));
	else
		tmp = Float64(Float64(l_m * sqrt(2.0)) * sqrt(Float64(Float64(n * U) * fma(U_42_, Float64(n / Float64(Om * Om)), Float64(-2.0 / Om)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[l$95$m, 1.85e-28], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.8e+95], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t - N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / Om), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(l$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(n * U), $MachinePrecision] * N[(U$42$ * N[(n / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.8500000000000001e-28

   1. Initial program 48.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 44.1

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 44.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 49.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 52.2%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 57.4

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 57.4%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if 1.8500000000000001e-28 < l < 1.79999999999999989e95

   1. Initial program 45.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(2 \cdot \frac{{\ell}^{2}}{Om} + \frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{{Om}^{2}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      3. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{\color{blue}{Om \cdot Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om}} + 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} + \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \frac{{\ell}^{2}}{Om}\right)\right)} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - -2 \cdot \frac{{\ell}^{2}}{Om}\right)}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om}}{Om} - \color{blue}{\frac{-2 \cdot {\ell}^{2}}{Om}}\right)\right)} \]

      8. div-sub N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\frac{\frac{{\ell}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)}{Om} - -2 \cdot {\ell}^{2}}{Om}}\right)} \]

   5. Simplified 51.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}} \]

   if 1.79999999999999989e95 < l

   1. Initial program 27.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot n\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      11. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\frac{\ell}{Om}}, \frac{\ell}{Om} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om} \cdot n}, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \color{blue}{\frac{\ell}{Om}} \cdot n, t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right)} \]

      16. sub-neg N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{t + \left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right)}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{\ell \cdot \ell}{Om}\right)\right) + t}\right)} \]

   4. Applied egg-rr 39.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)}} \]

   5. Taylor expanded in U around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   6. Step-by-step derivation

   7. Simplified 38.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{U*} \cdot \frac{\ell}{Om}, \frac{\ell}{Om} \cdot n, \mathsf{fma}\left(-2 \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + t\right) + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(-2 \cdot \ell\right) \cdot \frac{\ell}{Om} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{-2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2} + \left(t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\ell \cdot \ell}{Om}}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \ell}}{Om}, -2, t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, \color{blue}{t + \left(U* \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{U* \cdot \ell}{Om}} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{U* \cdot \ell}{Om} \cdot \color{blue}{\frac{\ell \cdot n}{Om}}\right)} \]

      12. frac-times N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \color{blue}{\frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}}\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\color{blue}{\left(U* \cdot \ell\right)} \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \color{blue}{\left(\ell \cdot n\right)}}{Om \cdot Om}\right)} \]

      17. *-lowering-*.f64 20.3

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{\color{blue}{Om \cdot Om}}\right)} \]

   9. Applied egg-rr 20.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t + \frac{\left(U* \cdot \ell\right) \cdot \left(\ell \cdot n\right)}{Om \cdot Om}\right)}} \]

   10. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \ell\right) \cdot \sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\sqrt{U \cdot \left(n \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)\right)}} \]

       7. associate-*r* N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right) \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\color{blue}{\left(U \cdot n\right)} \cdot \left(\frac{U* \cdot n}{{Om}^{2}} - 2 \cdot \frac{1}{Om}\right)} \]

       10. sub-neg N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\left(\frac{U* \cdot n}{{Om}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)}} \]

       11. associate-/l* N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \left(\color{blue}{U* \cdot \frac{n}{{Om}^{2}}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)\right)} \]

       12. accelerator-lowering-fma.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \color{blue}{\mathsf{fma}\left(U*, \frac{n}{{Om}^{2}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)}} \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \color{blue}{\frac{n}{{Om}^{2}}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{\color{blue}{Om \cdot Om}}, \mathsf{neg}\left(2 \cdot \frac{1}{Om}\right)\right)} \]

       16. associate-*r/ N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{Om}}\right)\right)} \]

       17. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \mathsf{neg}\left(\frac{\color{blue}{2}}{Om}\right)\right)} \]

       18. distribute-neg-frac N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{\mathsf{neg}\left(2\right)}{Om}}\right)} \]

       19. metadata-eval N/A

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{\color{blue}{-2}}{Om}\right)} \]

       20. /-lowering-/.f64 59.8

           \[\leadsto \left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \color{blue}{\frac{-2}{Om}}\right)} \]

   12. Simplified 59.8%

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\left(U \cdot n\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t - \frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om}, 2\right)}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\left(n \cdot U\right) \cdot \mathsf{fma}\left(U*, \frac{n}{Om \cdot Om}, \frac{-2}{Om}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{l\_m}{Om}, \frac{n \cdot l\_m}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -3e-153)
   (sqrt (* U (* (fma (/ l_m Om) (* (/ (* n l_m) Om) (- U* U)) t) (* 2.0 n))))
   (if (<= n 1.75e-102)
     (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
     (sqrt
      (*
       (* U (* 2.0 n))
       (+ t (* (- U* U) (* n (* l_m (/ l_m (* Om Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -3e-153) {
		tmp = sqrt((U * (fma((l_m / Om), (((n * l_m) / Om) * (U_42_ - U)), t) * (2.0 * n))));
	} else if (n <= 1.75e-102) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = sqrt(((U * (2.0 * n)) * (t + ((U_42_ - U) * (n * (l_m * (l_m / (Om * Om))))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -3e-153)
		tmp = sqrt(Float64(U * Float64(fma(Float64(l_m / Om), Float64(Float64(Float64(n * l_m) / Om) * Float64(U_42_ - U)), t) * Float64(2.0 * n))));
	elseif (n <= 1.75e-102)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t + Float64(Float64(U_42_ - U) * Float64(n * Float64(l_m * Float64(l_m / Float64(Om * Om))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -3e-153], N[Sqrt[N[(U * N[(N[(N[(l$95$m / Om), $MachinePrecision] * N[(N[(N[(n * l$95$m), $MachinePrecision] / Om), $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.75e-102], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3e-153

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 46.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 46.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 49.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 50.2%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. times-frac N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell \cdot n}{Om}\right)} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\ell}{Om} \cdot \left(\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\frac{\ell}{Om}}, \frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \color{blue}{\frac{\ell \cdot n}{Om}} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\color{blue}{\ell \cdot n}}{Om} \cdot \left(\mathsf{neg}\left(\left(U - U*\right)\right)\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. --lowering--.f64 57.3

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\color{blue}{\left(U - U*\right)}\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 57.3%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\frac{\ell}{Om}, \frac{\ell \cdot n}{Om} \cdot \left(-\left(U - U*\right)\right), t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if -3e-153 < n < 1.74999999999999993e-102

   1. Initial program 35.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 42.8

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 42.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 46.8

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 46.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 59.4

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 59.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]

   if 1.74999999999999993e-102 < n

   1. Initial program 45.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 44.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 44.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 51.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{U \cdot \left(\mathsf{fma}\left(\frac{\ell}{Om}, \frac{n \cdot \ell}{Om} \cdot \left(U* - U\right), t\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(l\_m \cdot \left(U* - U\right), \frac{n \cdot l\_m}{Om \cdot Om}, t\right)\right)}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(l\_m \cdot \frac{l\_m}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (if (<= n -7.5e-55)
   (sqrt (* U (* (* 2.0 n) (fma (* l_m (- U* U)) (/ (* n l_m) (* Om Om)) t))))
   (if (<= n 1.15e-103)
     (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
     (sqrt
      (*
       (* U (* 2.0 n))
       (+ t (* (- U* U) (* n (* l_m (/ l_m (* Om Om)))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double tmp;
	if (n <= -7.5e-55) {
		tmp = sqrt((U * ((2.0 * n) * fma((l_m * (U_42_ - U)), ((n * l_m) / (Om * Om)), t))));
	} else if (n <= 1.15e-103) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = sqrt(((U * (2.0 * n)) * (t + ((U_42_ - U) * (n * (l_m * (l_m / (Om * Om))))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	tmp = 0.0
	if (n <= -7.5e-55)
		tmp = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(l_m * Float64(U_42_ - U)), Float64(Float64(n * l_m) / Float64(Om * Om)), t))));
	elseif (n <= 1.15e-103)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t + Float64(Float64(U_42_ - U) * Float64(n * Float64(l_m * Float64(l_m / Float64(Om * Om))))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := If[LessEqual[n, -7.5e-55], N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(l$95$m * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision] * N[(N[(n * l$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 1.15e-103], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.50000000000000023e-55

   1. Initial program 48.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 46.5

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 46.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 50.5%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 48.5%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       2. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \color{blue}{\left(\ell \cdot \frac{\ell \cdot n}{Om \cdot Om}\right)} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       3. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \ell\right) \cdot \frac{\ell \cdot n}{Om \cdot Om}} + t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \ell, \frac{\ell \cdot n}{Om \cdot Om}, t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \ell}, \frac{\ell \cdot n}{Om \cdot Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right)} \cdot \ell, \frac{\ell \cdot n}{Om \cdot Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       7. --lowering--.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(U - U*\right)}\right)\right) \cdot \ell, \frac{\ell \cdot n}{Om \cdot Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \ell, \color{blue}{\frac{\ell \cdot n}{Om \cdot Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{fma}\left(\left(\mathsf{neg}\left(\left(U - U*\right)\right)\right) \cdot \ell, \frac{\color{blue}{\ell \cdot n}}{Om \cdot Om}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

       10. *-lowering-*.f64 51.7

           \[\leadsto \sqrt{\left(\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \ell, \frac{\ell \cdot n}{\color{blue}{Om \cdot Om}}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   12. Applied egg-rr 51.7%

       \[\leadsto \sqrt{\left(\color{blue}{\mathsf{fma}\left(\left(-\left(U - U*\right)\right) \cdot \ell, \frac{\ell \cdot n}{Om \cdot Om}, t\right)} \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if -7.50000000000000023e-55 < n < 1.15e-103

   1. Initial program 39.8%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 46.5

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 46.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 49.7

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 49.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 60.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 60.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]

   if 1.15e-103 < n

   1. Initial program 45.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 44.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 44.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 51.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7.5 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\ell \cdot \left(U* - U\right), \frac{n \cdot \ell}{Om \cdot Om}, t\right)\right)}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 54.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{l\_m}{Om \cdot Om}\\ t_2 := U \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;n \leq -6.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t + \left(n \cdot l\_m\right) \cdot \left(\left(U* - U\right) \cdot t\_1\right)\right)}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_2 \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(l\_m \cdot t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1 (/ l_m (* Om Om))) (t_2 (* U (* 2.0 n))))
   (if (<= n -6.2e-60)
     (sqrt (* t_2 (+ t (* (* n l_m) (* (- U* U) t_1)))))
     (if (<= n 9e-104)
       (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
       (sqrt (* t_2 (+ t (* (- U* U) (* n (* l_m t_1))))))))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = l_m / (Om * Om);
	double t_2 = U * (2.0 * n);
	double tmp;
	if (n <= -6.2e-60) {
		tmp = sqrt((t_2 * (t + ((n * l_m) * ((U_42_ - U) * t_1)))));
	} else if (n <= 9e-104) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = sqrt((t_2 * (t + ((U_42_ - U) * (n * (l_m * t_1))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = Float64(l_m / Float64(Om * Om))
	t_2 = Float64(U * Float64(2.0 * n))
	tmp = 0.0
	if (n <= -6.2e-60)
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(n * l_m) * Float64(Float64(U_42_ - U) * t_1)))));
	elseif (n <= 9e-104)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = sqrt(Float64(t_2 * Float64(t + Float64(Float64(U_42_ - U) * Float64(n * Float64(l_m * t_1))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.2e-60], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(n * l$95$m), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 9e-104], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(t + N[(N[(U$42$ - U), $MachinePrecision] * N[(n * N[(l$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.19999999999999976e-60

   1. Initial program 49.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 47.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 47.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 51.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om \cdot Om}\right)} \cdot \left(U - U*\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right)} \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right)} \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \color{blue}{\left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\color{blue}{\frac{\ell}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\frac{\ell}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)} \]

      9. --lowering--.f64 51.2

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \color{blue}{\left(U - U*\right)}\right)\right)} \]

   10. Applied egg-rr 51.2%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

   if -6.19999999999999976e-60 < n < 8.9999999999999995e-104

   1. Initial program 39.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 46.0

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 49.2

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 49.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 59.6

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 59.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]

   if 8.9999999999999995e-104 < n

   1. Initial program 45.9%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 44.3

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 44.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 51.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.2 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(n \cdot \ell\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(U* - U\right) \cdot \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(n \cdot l\_m\right) \cdot \left(\left(U* - U\right) \cdot \frac{l\_m}{Om \cdot Om}\right)\right)}\\ \mathbf{if}\;n \leq -1.75 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* U (* 2.0 n))
           (+ t (* (* n l_m) (* (- U* U) (/ l_m (* Om Om)))))))))
   (if (<= n -1.75e-60)
     t_1
     (if (<= n 4.2e-105)
       (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt(((U * (2.0 * n)) * (t + ((n * l_m) * ((U_42_ - U) * (l_m / (Om * Om)))))));
	double tmp;
	if (n <= -1.75e-60) {
		tmp = t_1;
	} else if (n <= 4.2e-105) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(Float64(U * Float64(2.0 * n)) * Float64(t + Float64(Float64(n * l_m) * Float64(Float64(U_42_ - U) * Float64(l_m / Float64(Om * Om)))))))
	tmp = 0.0
	if (n <= -1.75e-60)
		tmp = t_1;
	elseif (n <= 4.2e-105)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(U * N[(2.0 * n), $MachinePrecision]), $MachinePrecision] * N[(t + N[(N[(n * l$95$m), $MachinePrecision] * N[(N[(U$42$ - U), $MachinePrecision] * N[(l$95$m / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.75e-60], t$95$1, If[LessEqual[n, 4.2e-105], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.74999999999999988e-60 or 4.2e-105 < n

   1. Initial program 47.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 45.7

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 45.7%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 51.1%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om \cdot Om}\right)} \cdot \left(U - U*\right)\right)} \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(n \cdot \ell\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right)} \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right)} \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \color{blue}{\left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\color{blue}{\frac{\ell}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\frac{\ell}{\color{blue}{Om \cdot Om}} \cdot \left(U - U*\right)\right)\right)} \]

      9. --lowering--.f64 50.7

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \color{blue}{\left(U - U*\right)}\right)\right)} \]

   10. Applied egg-rr 50.7%

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \color{blue}{\left(\ell \cdot n\right) \cdot \left(\frac{\ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)}\right)} \]

   if -1.74999999999999988e-60 < n < 4.2e-105

   1. Initial program 39.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 46.0

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 46.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 49.2

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 49.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 59.6

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 59.6%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.75 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(n \cdot \ell\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om \cdot Om}\right)\right)}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(t + \left(n \cdot \ell\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om \cdot Om}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{l\_m \cdot \left(n \cdot l\_m\right)}{Om \cdot Om}, U*, t\right)\right)}\\ \mathbf{if}\;U* \leq -2.45 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;U* \leq 45:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(l\_m \cdot \frac{U \cdot -4}{Om}, n \cdot l\_m, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (* U (* (* 2.0 n) (fma (/ (* l_m (* n l_m)) (* Om Om)) U* t))))))
   (if (<= U* -2.45e+82)
     t_1
     (if (<= U* 45.0)
       (sqrt (fma (* l_m (/ (* U -4.0) Om)) (* n l_m) (* U (* t (* 2.0 n)))))
       t_1))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	double t_1 = sqrt((U * ((2.0 * n) * fma(((l_m * (n * l_m)) / (Om * Om)), U_42_, t))));
	double tmp;
	if (U_42_ <= -2.45e+82) {
		tmp = t_1;
	} else if (U_42_ <= 45.0) {
		tmp = sqrt(fma((l_m * ((U * -4.0) / Om)), (n * l_m), (U * (t * (2.0 * n)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	t_1 = sqrt(Float64(U * Float64(Float64(2.0 * n) * fma(Float64(Float64(l_m * Float64(n * l_m)) / Float64(Om * Om)), U_42_, t))))
	tmp = 0.0
	if (U_42_ <= -2.45e+82)
		tmp = t_1;
	elseif (U_42_ <= 45.0)
		tmp = sqrt(fma(Float64(l_m * Float64(Float64(U * -4.0) / Om)), Float64(n * l_m), Float64(U * Float64(t * Float64(2.0 * n)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(U * N[(N[(2.0 * n), $MachinePrecision] * N[(N[(N[(l$95$m * N[(n * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * U$42$ + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U$42$, -2.45e+82], t$95$1, If[LessEqual[U$42$, 45.0], N[Sqrt[N[(N[(l$95$m * N[(N[(U * -4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(n * l$95$m), $MachinePrecision] + N[(U * N[(t * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if U* < -2.45e82 or 45 < U*

   1. Initial program 44.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\frac{{\ell}^{2}}{{Om}^{2}}}\right) \cdot \left(U - U*\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}}\right) \cdot \left(U - U*\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{{Om}^{2}}\right)}\right) \cdot \left(U - U*\right)\right)} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \color{blue}{\frac{\ell}{{Om}^{2}}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

      6. *-lowering-*.f64 41.6

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \left(\ell \cdot \frac{\ell}{\color{blue}{Om \cdot Om}}\right)\right) \cdot \left(U - U*\right)\right)} \]

   5. Simplified 41.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)}\right) \cdot \left(U - U*\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   7. Step-by-step derivation

   8. Simplified 50.2%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\color{blue}{t} - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(t - \left(n \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}} \]

   10. Applied egg-rr 53.2%

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, -\left(U - U*\right), t\right) \cdot \left(n \cdot 2\right)\right) \cdot U}} \]

   11. Taylor expanded in U around 0

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \color{blue}{U*}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]
   12. Step-by-step derivation

   13. Simplified 53.2%

       \[\leadsto \sqrt{\left(\mathsf{fma}\left(\frac{\ell \cdot \left(\ell \cdot n\right)}{Om \cdot Om}, \color{blue}{U*}, t\right) \cdot \left(n \cdot 2\right)\right) \cdot U} \]

   if -2.45e82 < U* < 45

   1. Initial program 44.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{-4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om} + 2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right) + -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, \color{blue}{U \cdot \left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \color{blue}{\left(n \cdot t\right)}, -4 \cdot \frac{U \cdot \left({\ell}^{2} \cdot n\right)}{Om}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \color{blue}{\frac{-4 \cdot \left(U \cdot \left({\ell}^{2} \cdot n\right)\right)}{Om}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right) \cdot -4}}{Om}\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\color{blue}{\left(U \cdot \left({\ell}^{2} \cdot n\right)\right)} \cdot -4}{Om}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \color{blue}{\left({\ell}^{2} \cdot n\right)}\right) \cdot -4}{Om}\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      12. *-lowering-*.f64 38.9

          \[\leadsto \sqrt{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot n\right)\right) \cdot -4}{Om}\right)} \]

   5. Simplified 38.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, U \cdot \left(n \cdot t\right), \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \left(n \cdot t\right)\right) \cdot 2} + \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(U \cdot \left(n \cdot t\right), 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(n \cdot t\right) \cdot U}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      4. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{n \cdot \left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \color{blue}{\left(t \cdot U\right)}, 2, \frac{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot -4}{Om}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(U \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)\right) \cdot \frac{-4}{Om}}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot U\right)} \cdot \frac{-4}{Om}\right)} \]

      9. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right)} \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \color{blue}{\left(\ell \cdot n\right)}\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \color{blue}{\left(U \cdot \frac{-4}{Om}\right)}\right)} \]

      15. /-lowering-/.f64 46.9

          \[\leadsto \sqrt{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \color{blue}{\frac{-4}{Om}}\right)\right)} \]

   7. Applied egg-rr 46.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(n \cdot \left(t \cdot U\right), 2, \left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\ell \cdot \left(\ell \cdot n\right)\right) \cdot \left(U \cdot \frac{-4}{Om}\right) + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \left(\ell \cdot \left(\ell \cdot n\right)\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell\right) \cdot \left(\ell \cdot n\right)} + \left(n \cdot \left(t \cdot U\right)\right) \cdot 2} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(U \cdot \frac{-4}{Om}\right) \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(U \cdot \frac{-4}{Om}\right) \cdot \ell}, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{U \cdot -4}{Om}} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\color{blue}{U \cdot -4}}{Om} \cdot \ell, \ell \cdot n, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \color{blue}{\ell \cdot n}, \left(n \cdot \left(t \cdot U\right)\right) \cdot 2\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{2 \cdot \left(n \cdot \left(t \cdot U\right)\right)}\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(2 \cdot n\right) \cdot \left(t \cdot U\right)}\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot t\right)}\right)} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{\left(U \cdot \left(2 \cdot n\right)\right)} \cdot t\right)} \]

      15. associate-*l* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, \color{blue}{U \cdot \left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \color{blue}{\left(\left(2 \cdot n\right) \cdot t\right)}\right)} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

      19. *-lowering-*.f64 51.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\color{blue}{\left(n \cdot 2\right)} \cdot t\right)\right)} \]

   9. Applied egg-rr 51.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot -4}{Om} \cdot \ell, \ell \cdot n, U \cdot \left(\left(n \cdot 2\right) \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U* \leq -2.45 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om \cdot Om}, U*, t\right)\right)}\\ \mathbf{elif}\;U* \leq 45:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell \cdot \frac{U \cdot -4}{Om}, n \cdot \ell, U \cdot \left(t \cdot \left(2 \cdot n\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \mathsf{fma}\left(\frac{\ell \cdot \left(n \cdot \ell\right)}{Om \cdot Om}, U*, t\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 35.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (n U t l_m Om U*) :precision binary64 (sqrt (* (* 2.0 U) (* n t))))

C

l_m = fabs(l);
double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return sqrt(((2.0 * U) * (n * t)));
}

Fortran

l_m = abs(l)
real(8) function code(n, u, t, l_m, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((2.0d0 * u) * (n * t)))
end function

Java

l_m = Math.abs(l);
public static double code(double n, double U, double t, double l_m, double Om, double U_42_) {
	return Math.sqrt(((2.0 * U) * (n * t)));
}

Python

l_m = math.fabs(l)
def code(n, U, t, l_m, Om, U_42_):
	return math.sqrt(((2.0 * U) * (n * t)))

Julia

l_m = abs(l)
function code(n, U, t, l_m, Om, U_42_)
	return sqrt(Float64(Float64(2.0 * U) * Float64(n * t)))
end

MATLAB

l_m = abs(l);
function tmp = code(n, U, t, l_m, Om, U_42_)
	tmp = sqrt(((2.0 * U) * (n * t)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[n_, U_, t_, l$95$m_, Om_, U$42$_] := N[Sqrt[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 44.4%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right)} \cdot \left(n \cdot t\right)} \]

   4. *-lowering-*.f64 35.2

      \[\leadsto \sqrt{\left(2 \cdot U\right) \cdot \color{blue}{\left(n \cdot t\right)}} \]

5. Simplified 35.2%

   \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))